Engine · Music and the beat

Beat Engine

Eight spheres and a black hole that dance to any song. A trained eight-band detector hears kick, bass, snare, toms, voice, keys, hat and air; the bass pounds and pulls the pieces apart, the mids turn and twist, the treble flashes. Press Sync, play a song, or record a take.

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Engine 00 · Fractals, attractors and light

Mandelbox Amazing Family Engine

Mandelbox, Amazing Surf, Surf Mod1 and Box Julia attractors, ray-marched on the GPU. Each fractal is mapped: its real extent sets the render window and export range. Six template finishes, an iridescent switch, solid or glass, and seven motions.

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Engine 00 · Fractals, folds and light

queueBIT Fractal Engine

A Menger sponge folded by box, sphere, lattice and tetrahedral transforms, ray-marched on the GPU, with an optional fourth dimension. Seven motions, the template palettes and an iridescent switch.

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Engine 01 · Perception, waves and collapse

ψcience Perception Engine

Overlapping wave candidates pass through seven perception lenses, from Weber-Fechner compression to projective and process transforms, and collapse when the entropy drive and threshold say so. Presets, wave classes, and a live entropy readout.

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Engine 01 · Mathematics and computer graphics

Fibonacci Squares Engine

Spirals, towers, sunflowers and the platonic solids, built from Fibonacci numbers and the golden ratio. Wire, glass, solid and iridescent finishes, with still and animated GLB export.

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Engine 02 · Mathematics, botany and design

Pythagorean Tree Engine

Trees grown from a² + b² = c²: every branch splits into two legs. Flat and 360° styles, a golden-angle preset, spiral twist, palette colours and twist animation for GLB export.

Open the engine Where the maths is

Engine 03 · Mathematics, textiles and design

Sphere & Torus Weave Engine

Warp and weft wires woven over spheres, tori, cylinders and Möbius bands, with plain, twill, rib, basket, satin and hex patterns. Wire, glass and solid finishes, an iridescent wrapper, and a spinning GLB export.

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UUON

Mathematical tools you can use in your own projects.

Tools built from mathematics: shrinking data, drawing fractal shapes, and simulating waves and orbits. Use them through a web address (an API), a code package (an SDK), or an AI assistant (MCP). Every service below is checked from this page.

Explore the APIs Install an SDK

Checking services…

The two engines

What each engine is, and where its maths lives

Both run in the browser, export to GLB for any 3D app, and share one template: the same finish, colour, and control rules.

Fibonacci Squares Engine v6 · Mathematics, computer graphics

Fibonacci squares, spirals and the golden ratio

Builds forms from Fibonacci numbers and the golden ratio: spiral towers, stairs, sunflower seed heads, and the platonic solids.

Special characteristics

Eleven forms: octopus arms, clusters, cube, tetra, octa, dodeca, icosa, golden sphere, golden spiral, golden sunflower, Fibonacci stair.
Wire, glass and solid finishes, with an iridescent shader that colours by position.
Live controls for Fibonacci terms, growth ratio, spiral twist and cube size; full and simple views.
Exports a still GLB or a 4-second animated GLB, with colours baked from what you see.

Where the maths is implemented

Field	Idea	Where in the engine
Number theory	Fibonacci sequence 1, 1, 2, 3, 5…	`fib()`, `fibSquares()`
Geometry	Golden ratio φ ≈ 1.618 and the growth ratio	growth-ratio control, `PHI`
Botany	Golden angle 137.5° (2.39996 rad)	golden sunflower and golden sphere placement
Spirals	Golden spiral arcs and the rising spiral tower	`fibArcs()`, spiral branch in `update()`
Polyhedra	Platonic solids: tetra, octa, dodeca, icosa	shape table `SH`
Shading	Cosine colour palette for iridescence	vertex and fragment shaders `vs`, `fs`

Pythagorean Tree Engine · Mathematics, botany, design

Pythagorean trees from a² + b² = c²

Each branch is the hypotenuse of a right triangle; its two children are the legs. Repeat the split and a tree grows.

Special characteristics

Flat (Pythagorean) and 360° fan styles, with a golden-angle preset at 31.72°.
Ten presets: classic, balanced and unbalanced squares, line segment, n-ary (3, 4, 5), L-system and wide spread.
Angle, length, girth and spiral twist glide on springs; depth and branch count rebuild the tree.
Design-palette colours, iridescent as a filter, and wire, glass and solid finishes.
Exports GLB (still or twist animation), OBJ, and a graph of the tree's nodes and metrics.

Where the maths is implemented

Field	Idea	Where in the engine
Geometry	Pythagorean theorem a² + b² = c²: a branch splits into two legs	`layoutTree()`, `childDirs()`
Botany	Golden angle 31.72° and 137.5° phase in 360° mode	golden button, `childDirs()` phase term
Rotation	Spiral twist: each level turns about the trunk	`childDirs()` twist term
Dynamics	Damped spring, stiffness 120, damping 16	`stepTree()`
Quaternions	Turning and twisting motion, baked per frame	`exportTreeGLB()` tracks
Graph metrics	Node depth, length, surface area and volume per branch	`downloadGraph()`

Sphere & Torus Weave Engine · Mathematics, textiles, design

Woven surfaces on spheres, tori and bands

Warp and weft wires are laid over a parametric surface. Each crossing follows a weave pattern, so the surface reads as cloth.

Special characteristics

Four surfaces: sphere, torus, cylinder and Möbius band, each with its own parameter grid.
Six weave patterns: plain, twill, rib, basket, satin and hex (with nested layers).
Wire, glass and solid finishes, with an iridescent thin-film wrapper and a colour hue.
Exports a textured PNG, a static OBJ, and a spinning GLB with glass transparency kept.

Where the maths is implemented

Field	Idea	Where in the engine
Parametric surfaces	Sphere, torus, cylinder and Möbius parametrisation	`Sf(u, v, o)`
Combinatorics	Over and under sign for each crossing, per weave pattern	`sg(i, j)`
Smooth profiles	Cosine easing between crossings	`ci(a, b, f)`
Hexagonal packing	Hex cells on a latitude grid, with nested layers	hex branch of `build()`
Thin-film colour	Cosine palette shifted by view angle (iridescence)	wrapper fragment shader in `irMat()`

More apps

Parsir, Quantum, Weave and wave views

Live waveRealistic oceanClick to open full screen

FourierPlucked stringClick to open full screen

ParsirAlgorithmic EngineOpen the app

QuantumWave VisualizerOpen the app

Services

What UUON offers, and where to find it

Each service has its own address, and its status is shown live.

uuon.world— Gate API Paid algorithm engines: IFS codecs, fractal renders and analysis.

clouud.uuon.world— Clouud Agent Free compression-field endpoints, benchmarks and registry search.

census.uuon.world— Census Governor The knowledge census: twelve houses under a versioned truth rule.

arcaide.pages.devOpen ArcAIDE Interactive mathematical engines, learned by our agents before public release.

by invitationPrivate Business MCP Model Context Protocol tools for agents, access-controlled.

npmjs.com/@uuon-foundation3 packages npm SDKs Three engine packages under @uuon-foundation.

UUON Möbius Engine · non-orientable P-surface

01 · APIs

Ready-to-use web services

Requests and answers in a standard data format. Some tools are free; the advanced math is paid per use.

Gate API

Paid algorithm engines: IFS codecs, L-system and parametric IFS, Mandelbulb and Menger-Kleinian renders, fractal analysis.

—

Base URL https://uuon.world

Gate API documentation →

Engines

POST/algorithms/ifs-inverse-codec/encode
POST/algorithms/lsystem-ifs/generate
POST/algorithms/parametric-ifs/discover
POST/algorithms/mandelbulb/render
POST/algorithms/fractal-analysis/analyze

Authenticate with an x-api-key header. Free compression endpoints are on the Clouud Agent below.

Access

GET/auth/githubGitHub sign-in
POST/auth/walletWallet sign-in
POST/registerAPI key
GET/billing/pricingLive price list

Paid engines · loading live prices

Engine	Price	Unit

Clouud Agent

Free compression-field endpoints, benchmarks, registry search and house lookups.

—

Base URL https://clouud.uuon.world

Clouud documentation →

POST/algorithms/compression-field/compress
POST/algorithms/compression-field/entropy
POST/benchmark
GET/api/registry/search
GET/api/registry/count
GET/api/house/:id

Example request

curl -X POST https://clouud.uuon.world/algorithms/compression-field/compress \
  -H "content-type: application/json" \
  -d '{"data": [1, 2, 3, 1, 2, 3, 1, 2, 3]}'

Census Governor

The knowledge census: twelve houses, from Mathematics to General Intelligence & Integration, under a versioned truth rule.

—

Base URL https://census.uuon.world

Census documentation →

GET/Houses and truth-rule version

02 · SDKs

Code packages you can install

Free packages, shared on npm (a public library of code). Names start with @uuon-foundation. Licensed under USAL-1.0, see License below.

wave-field-3d-engine

3D wave-field simulation and rendering.

npm i @uuon-foundation/wave-field-3d-engine

Live demo ↗ npm ↗

pythagorean-graph-engine

Pythagorean tree and graph generation.

npm i @uuon-foundation/pythagorean-graph-engine

Live demo ↗ npm ↗

pscience-perception-engine

Perception-science engine.

npm i @uuon-foundation/pscience-perception-engine

Live demo ↗ npm ↗

Source for public engines: github.com/UUON-Foundation ↗

Zenith / zɛnɪθ by UUON Foundation on Sketchfab

Toroidal Energy by UUON Foundation on Sketchfab

03 · MCP

Tools for AI assistants

Connectors that let AI assistants use UUON services directly.

Business MCP

Business tools for AI assistants. Only approved users can connect, and each connection is checked by a sign-in gate.

By invitation

Request access: support@uuon.world

04 · Engines

Engines

Live engines that run in your browser. More interactive tools are on ArcAIDE.

Mandelbox Amazing Family

Mandelbox, Amazing Surf and Box Julia attractors by GLSL ray marching.

Open engine →

Beat Engine

Eight spheres and a black hole that dance to any song, by GLSL ray marching.

Open engine →

queueBIT Fractal Engine

Folded Menger sponge by GLSL ray marching, with a fourth dimension.

Open engine →

Wave Field 3D Engine

24 wave algorithms on 3D geometry, with colour by live displacement and animated GLB export.

Open engine →

Pythagorean Graph Engine

Recursive tree topology from Pythagorean triples, with colour ramps and full graph JSON export.

Open engine →

ψcience Perception Engine

Competing interpretations as a live interference field, shown in colour.

Open engine →

Parametric Solids Engine

Ten parametric solids with closed-form area and volume, a colour fill and public metadata JSON.

Open engine →

Compression Field Engine

Four-zone infall model rendering information encoding as a live particle field.

Open engine →

Parametric Geometry Engine

The Möbius strip with five controls, the exact parametric equation, live area and length, and tracking over time.

Open engine →

Recursive Fractal Engine

7,744 dynamical systems from one compiled shader, with quantum probability landscapes.

Open engine →

Learned by doing. Each engine lets you change a value and watch the result, so the mathematics is understood by using it, not only by reading about it.
Checked before it is listed. Every engine is measured by an automated observer, so what you see has been tested, not just demonstrated.
Tested by agents first. Our agents get access before people do. They learn the engines and report what works, so the public release is built on that evidence.
Nothing to install. The engines run in the browser, on any device that can open this page.

UUON Möbius Engine · non-orientable P-surface

Models

3D shapes we made

Fractal and many-sided shapes, shared on Sketchfab (a 3D model site). Each shows a picture until you open the 3D view.

White Hole

View on Sketchfab ↗

Recursive Kleinian Scaffold

View on Sketchfab ↗

Null Set Empty - Δmension M.U:

View on Sketchfab ↗

Cuboctahedron Qubit

View on Sketchfab ↗

Torus Catenoid Enneper - Warp Tech.

View on Sketchfab ↗

uuon_ifs_none (6)_obj

View on Sketchfab ↗

uuon_ifs_box (7)

View on Sketchfab ↗

uuon_ifs_box_hd

View on Sketchfab ↗

05 · CENS

1,000 free CENS credits for the first 100 builders

CENS are credits you spend to use UUON's tools. They are not cryptocurrency, not an investment, and cannot be transferred.

— of 100 claimed

1,000CENS per verified account
1claim per wallet or GitHub account
10 CENS= $1 of engine usage

Signing in asks your wallet to sign a message (EIP-4361). It sends no transaction and costs no gas. Prefer GitHub? Sign in with GitHub to get an API key, then POST /cens/claim with x-api-key.

Spend CENS on any Gate engine by adding the header X-Pay-With: cens. See Gate API docs.

Buy CENS — every pack includes a 50% bonus

Sign in with your wallet, then pay on Stripe's secure checkout page. CENS are service credits for UUON engines and APIs: not a cryptocurrency, not an investment, non-transferable. Refunded or disputed payments take back the unspent CENS.

06 · Status

Are our services working?

Checked from the internet each time you load this page (saved for one minute).

Service	State	Response
Checking…

Built with

Tools UUON Foundation builds with. Product names are trademarks of their owners; no partnership or endorsement is implied.

License

USAL-1.0

UUON Source Attribution License. Attribution is required on all derivatives, and use for AI training is not permitted. For commercial licensing, contact licensing@uuon.world.